8. Tensor Field Visualization

Transcription

1 8. Tensor Field Visualization Tensor: extension of concept of scalar and vector Tensor data for a tensor of level k is given by t i1,i2,,ik (x 1,,x n ) Second-order tensor often represented by matrix Examples: Diffusion tensor (from medical imaging, see later) Material properties (material sciences): Conductivity tensor Dielectric susceptibility Magnetic permutivity Stress tensor Diffusion Tensor Typical second-order tensor: diffusion tensor Diffusion: based on motion of fluid particles on microscopic level Probabilistic phenomenon Based on particle s Brownian motion Measurements by modern MR (magnetic resonance) scanners Diffusion tensor describes diffusion rate into different directions via symmetric tensor (probability density distribution) In 3D: representation via 3*3 symmetric matrix 2

2 8.1. Diffusion Tensor Symmetric diffusion matrix can be diagonalized: Real eigenvalues λ 1 λ 2 λ 3 Eigenvectors are perpendicular Isotropy / anisotropy: Spherical: λ 1 =λ 2 =λ 3 Linear: λ 2 λ 3 0 Planar: λ 1 λ 2 und λ Diffusion Tensor Arbitrary vectors are generally deflected after matrix multiplication Deflection into direction of principal eigenvector (largest eigenvalue) 4

3 8.2. Basic Mapping Techniques Matrix of images Slices through volume Each image shows one component of the matrix Basic Mapping Techniques Uniform grid of ellipsoids Second-order symmetric tensor mapped to ellipsoid Sliced volume [Pierpaoli et al. 1996] 6

4 8.2. Basic Mapping Techniques Uniform grid of ellipsoids Normalized sizes of the ellipsoids [Laidlaw et al. 1998] Basic Mapping Techniques Brushstrokes [Laidlaw et al. 1998] Influenced by paintings Multivalued data Scalar intensity Sampling rate Diffusion tensor Textured strokes scalar sampling rate tensor 8

5 8.2. Basic Mapping Techniques Ellipsoids in 3D Problems: Occlusion Missing continuity Basic Mapping Techniques Haber glyphs [Haber 1990] Rod and elliptical disk Better suited to visualize magnitudes of the tensor and principal axis 10

6 8.2. Basic Mapping Techniques Box glyphs [Johnson et al. 2001] Basic Mapping Techniques Reynolds glyph [Moore et al. 1994] 12

7 8.2. Basic Mapping Techniques Glyph for fourth-order tensor (wave propagation in crystals) Basic Mapping Techniques Generic iconic techniques for feature visualization [Post et al. 1995] 14

8 8.2. Basic Mapping Techniques Glyph probe for local flow field visualization [Leeuw, Wijk 1993] Arrow: particle path Green cap: tangential acceleration Orange ring: shear (with respect to gray ring) Hyperstreamlines and Tensorlines Hyperstreamlines [Delmarcelle, Hesselink 1992/93] Streamlines defined by eigenvectors Direction of streamline by major eigenvector Visualization of the vector field defined by major eigenvector Other eigenvectors define cross-section 16

9 8.4. Hyperstreamlines and Tensorlines Idea behind hyperstreamlines: Major eigenvector describes direction of diffusion with highest probability density Ambiguity for (nearly) isotropic case Hyperstreamlines and Tensorlines Problems of hyperstreamlines Ambiguity in (nearly) isotropic regions: Partial voluming effect, especially in low resolution images (MR images) Noise in data Solution: tensorlines Tensorline Hyperstreamline Arrows: major eigenvector 18

10 8.4. Hyperstreamlines and Tensorlines Tensorlines [Weinstein, Kindlmann 1999] Advection vector Stabilization of propagation by considering Input velocity vector Output velocity vector (after application of tensor operation) Vector along major eigenvector Weighting of three components depends on anisotropy at specific position: Linear anisotropy: only along major eigenvector Other cases: input or output vector Hyperstreamlines and Tensorlines Tensorlines 20

11 8.3. Hue-Balls and Lit-Tensors Hue-balls and lit-tensors [Kindlmann, Weinstein 1999] Ideas and elements Visualize anisotropy (relevant, e.g., in biological applications) Color coding Opacity function Illumination Volume rendering Hue-Balls and Lit-Tensors Color coding (hue-ball) Fixed, yet arbitrary input vector (e.g., user specified) Color coding for output vector Coding on sphere Idea: Deflection is strongly coupled with anisotropy 22

12 8.3. Hue-Balls and Lit-Tensors Barycentric opacity mapping Emphasize important features Make unimportant regions transparent Can define 3 barycentric coordinates c l, c p,c s Hue-Balls and Lit-Tensors Barycentric opacity mapping (cont.) Examples for transfer functions 24

13 8.3. Hue-Balls and Lit-Tensors Lit-tensors Similar to illuminated streamlines Illumination of tensor representations Provide information on direction and curvature Cases Linear anisotropy: same as illuminated streamlines Planar anisotropy: surface shading Other cases: smooth interpolation between these two extremes Hue-Balls and Lit-Tensors Lit-tensors (cont.) Example 26

14 8.3. Hue-Balls and Lit-Tensors Hue-Balls and Lit-Tensors Variation: streamtubes and streamsurfaces [Zhang et al. 2000] Streamtubes: linear anisotropic regions Streamsurfaces: planar anisotropic surfaces 28 linear planar

15 Visualization pipeline and classification visualization pipeline mapping classification sensors geometry: lines surfaces voxels attributes: color texture transparency simulation daten vis-data filter map renderable representations render visualization images videos interaction data bases 3D 2D 1D volume rend. isosurfaces height fields color coding stream ribbons topology arrows LIC glyphs icons attribute symbols scalar vector tensor/mv different grid types different algorithms 3D scalar fields cartesian medical datasets 3D vector fields un/structured CFD trees, graphs, tables, data bases InfoVis 29 Interactive Visualization of Huge Datasets CFD FE CT MR PET simulation sensors geometry: lines surfaces attributes: color structure images videos voxels transparency raw data visualization data renderable representation visualization steering filtering mapping rendering too much data too many cells too many triangles hierarchical representations mesh optimization feature extraction adaptive algorithms polygon reduction progressive techniques scene graphoptimization graphics hardware (i.e. textures) Optimization of all steps of the visualization pipeline Employ graphics hardware in rendering, mapping, and filtering interactions 30